A289369 Coefficients in expansion of (E_4^3/E_6^2)^(1/24).

1, 72, 11232, 5461344, 2029222656, 924074630640, 411487620614784, 192705317913673152, 91031590937141544960, 43814578627107100088424, 21291642032558036150652480, 10450287314646252538819378464, 5166676457072455262194208351232

Offset: 0

Author

Seiichi Manyama, Jul 04 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..367

Crossrefs

(E_4^3/E_6^2)^(k/288): A289365 (k=1), A299694 (k=2), A299696 (k=3), A299697 (k=4), A299698 (k=6), A299943 (k=8), A299949 (k=9), this sequence (k=12), A299950 (k=16), A299951 (k=18), A299953 (k=24), A299993 (k=32), A299994 (k=36), A300052 (k=48), A300053 (k=72), A300054 (k=96), A300055 (k=144), A289209 (k=288).

Cf. A108091 (E_4^(1/8)), A109817 (E_6^(1/12)).

Cf. A289367, A294978, A294979.

Programs

Mathematica

nmax = 20; CoefficientList[Series[((1 + 240*Sum[DivisorSigma[3,k]*x^k, {k, 1, nmax}])^3 / (1 - 504*Sum[DivisorSigma[5,k]*x^k, {k, 1, nmax}])^2)^(1/24), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 08 2017 *)

Formula

G.f.: Product_{n>=1} (1-q^n)^(-12*A289367(n)).

a(n) ~ c * exp(2*Pi*n) / n^(11/12), where c = 2^(1/3) * Pi^(1/4) / (3^(1/24) * Gamma(1/12) * Gamma(1/4)^(1/3)) = 0.0907014320494145997187363667820553893... - Vaclav Kotesovec, Jul 08 2017, updated Mar 04 2018

a(n) * A289368(n) ~ -(sqrt(3)-1) * exp(4*Pi*n) / (24*sqrt(2)*Pi*n^2). - Vaclav Kotesovec, Mar 04 2018